Question

Dadas as matrizes A= (0 2) (1 -1) e B= (bij) 2x2 seno bij= 2i- j para todo bij, determine a matriz x tal que:

X+ 2.B = -A

Answer 1

Vamos começar determinando a matriz B:

$B~=~\left[\begin{array}{ccc}b_{11}&b_{12}\\b_{21}&b_{22}\end{array}\right] \\\\\\B~=~\left[\begin{array}{ccc}2~.~1-1&2~.~1-2\\2~.~2-1&2~.~2-2\end{array}\right] \\\\\\B~=~\left[\begin{array}{ccc}1&0\\3&2\end{array}\right]$

Agora podemos montar a equação matricial:

$X~+~2.B~=~-A\\\\\\\left[\begin{array}{ccc}x_{11}&x_{12}\\x_{21}&x_{22}\end{array}\right]~+~2~.~\left[\begin{array}{ccc}1&0\\3&2\end{array}\right]~=~-\left[\begin{array}{ccc}0&2\\1&-1\end{array}\right]\\\\\\\\\left[\begin{array}{ccc}x_{11}&x_{12}\\x_{21}&x_{22}\end{array}\right]~+~\left[\begin{array}{ccc}2~.~1&2~.~0\\2~.~3&2~.~2\end{array}\right]~=~\left[\begin{array}{ccc}-0&-2\\-1&-(-1)\end{array}\right]$

$\left[\begin{array}{ccc}x_{11}&x_{12}\\x_{21}&x_{22}\end{array}\right]~+~\left[\begin{array}{ccc}2&0\\6&4\end{array}\right]~=~\left[\begin{array}{ccc}0&-2\\-1&1\end{array}\right]\\\\\\\\\left[\begin{array}{ccc}x_{11}&x_{12}\\x_{21}&x_{22}\end{array}\right]~=~\left[\begin{array}{ccc}0&-2\\-1&1\end{array}\right]~-~\left[\begin{array}{ccc}2&0\\6&4\end{array}\right]$

$\left[\begin{array}{ccc}x_{11}&x_{12}\\x_{21}&x_{22}\end{array}\right]~=~\left[\begin{array}{ccc}0-2&-2-0\\-1-6&1-4\end{array}\right]\\\\\\\\X~=~\left[\begin{array}{ccc}-2&-2\\-7&-3\end{array}\right]$